University of Wisconsin - Madison. Abstract. AMES-HET ASITP MADPH January 1999

Transcription

1 University of Wisconsin - Madison AMES-HET ASITP MADPH January 1999 NEUTRINO MIXING, CP/T VIOLATION AND TEXTURES IN FOUR-NEUTRINO MODELS V. Barger 1, Yuan-Ben Dai,K.Whisnant 3, and Bing-Lin Young 3 1 Department of Physics, University of Wisconsin, Madison, WI 53706, USA Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing , China 3 Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Abstract We examine the prospects for determining the neutrino mixing matrix and for observing CP and T violation in neutrino oscillations in four-neutrino models. We focus on a general class of four-neutrino models with two pairs of nearly degenerate mass eigenstates separated by approximately 1 ev, which can describe the solar, atmospheric and LSND neutrino data. We present a general parametrization of these models and discuss in detail the determination of the mixing parameters and the mass matrix texture from current and future neutrino data in the case where ν e and ν µ each mix primarily with one other neutrino. We find that measurable CP/T violation in long-baseline experiments, with amplitude at the level of the LSND signal, is possible given current experimental constraints. Also, additional oscillation effects in shortand long-baseline experiments may be measurable in many cases. We point out that, given separate scales for the mass-squared differences of the solar and atmospheric oscillations, observable CP/T violation effects in neutrino oscillations signals the existence of a sterile neutrino. We examine several textures of the neutrino mass matrix and determine which textures can have measurable CP/T violation in neutrino oscillations in long-baseline experiments. We also briefly discuss some possible origins of the neutrino mass terms in straightforward extensions of the Standard Model.

2 I. INTRODUCTION Our view of the neutrino sector of the Standard Model has recently undergone a revolutionary change. Observations of solar neutrinos [1 5], atmospheric neutrinos [6 8], and accelerator neutrinos [9] all indicate deviations from their predicted values in the Standard Model with massless neutrinos. In each case the observation can be understood in terms of neutrino oscillations which in turn requires nondegenerate neutrino masses. The accelerator evidence for oscillations is least secure, with preliminary data from the KARMEN Collaboration [10] excluding some regions of oscillation parameters preferred by the LSND data [9]. Since the solar, atmospheric, and LSND neutrino experiments have different L/E (the ratio of oscillation distance to neutrino energy), different orders of magnitude of neutrino masssquared differences δm are required to properly describe all features of the data [11]. This need for three small but distinct mass-squared differences naturally leads to the consideration of more than three light neutrino flavors. Any additional light neutrino must be sterile, i.e., without Standard Model gauge interactions, to be consistent with the well-established LEP measurements of Z ν ν [1]. From quite general arguments it has been shown [13,14] that a neutrino spectrum with two pairs of nearly degenerate mass eigenstates, separated by a gap of order 1 ev, is required to satisfy all of the constraints from solar, atmospheric, accelerator, and reactor data. Sterile neutrinos (which we we denote as ν s ) have long been considered as an option for neutrino oscillations [15,16]. More recently a number of models have been proposed that utilize one or more sterile neutrinos to describe the existing neutrino data [14,17 0] or to explain r-process nucleosynthesis [1]. However, if sterile neutrinos mix with active flavor neutrinos they may be stringently constrained by Big Bang nucleosynthesis (BBN). In standard BBN phenomenology, the mass-squared difference δm and the mixing angle between a sterile and active neutrino must satisfy the bound δm sin θ<10 7 ev, (1) to avoid thermal overpopulation of the extra, sterile neutrino species []. The restriction in Eq. (1) would appear to rule out all sterile-active mixing except for small-angle MSW or vacuum mixing of solar neutrinos. However, some recent estimates of N ν using higher inferred abundance of 4 He yield a considerably weaker bound than that given in Eq. (1) [3]. Thus BBN may still allow sizeable mixing between sterile and active neutrinos, so models with both small and large mixings with sterile neutrinos can be considered. In this paper we examine the phenomenological consequences of four-neutrino models in which there are two pairs of neutrinos with nearly degenerate mass eigenstates separated by about 1 ev, where the mass separations within the pairs are several orders of magnitude smaller. We begin with a general parametrization of the four-neutrino mixing matrix, and review the current experimental constraints. We then discuss the simple situation where ν e mixes dominantly with ν s or ν τ in solar neutrino oscillations and ν µ mixes dominantly with a fourth neutrino (ν τ or ν s ) in atmospheric neutrino oscillations. This situation, which we refer to as the dominant mixing scheme, has been shown to fit the existing data reasonably well. For dominant mixing we find that the neutrino mixing matrix can be effectively analyzed in terms of blocks, where the diagonal blocks can be approximated by simple twoneutrino rotations and the off-diagonal blocks are small but non-vanishing. We then study

3 the relationship between the phenomenology of neutrino oscillations with CP violation and the texture of the neutrino mass matrix in models where the two lightest states are much lighter than the two heaviest states. Since the neutrino mass matrix can in general be complex, and therefore lead to a mixing matrix with complex elements, CP violation can naturally arise in neutrino oscillations. The pertinent question is the size of the violation and how to observe it. We find that if CP violation exists, its size may be measurable, and has approximately the same amplitude as indicated by the LSND experiment. In many cases there are small amplitude ν e ν τ oscillations that may be measurable in either short- or long-baseline experiments. Furthermore, some also have small amplitude ν µ ν τ oscillations in short-baseline experiments. We discuss how oscillation measurements in solar, atmospheric, short- and long-baseline neutrino experiments can, in some cases, determine all but one of the four-neutrino mixing matrix parameters accessible to oscillation measurements. We also discuss the minimal Higgs boson spectrum needed to obtain the different types of four-neutrino mass matrices, and their consequences for CP violation. This paper is organized as follows. In Sec. II we present our parametrization for the four-neutrino mixing matrix and expressions for the oscillation probabilities in the case of two pairs of nearly degenerate mass eigenstates separated by about 1 ev. We discuss the ways in which CP violation, if it exists, may be observed, and the number of observable CP violation parameters. In Sec. III we summarize the current constraints on the four-neutrino mixing matrix and discuss in detail the implications of the dominant mixing scheme. We investigate the CP violation effects for several mass matrix textures in Sec. IV. In Sec. V we briefly discuss some of the consequences of neutrino mass for non-oscillation experiments, such as rare decays and charged lepton electric dipole moments, and we emphasize the importance in searching for these rare events, which can reveal new physics effects other than neutrino masses. In Sec. VI we summarize our results. Finally, in Appendix A we review the number of independent parameters in the mixing matrix of Majorana neutrinos, in Appendix B we discuss the modest extensions of the Standard Model Higgs sector that allow us to obtain the mass matrix textures, and in Appendix C we determine the neutrino mass spectrum and mixing matrix for a particular neutrino mass matrix with CP violation. II. OSCILLATION PROBABILITIES A. General Formalism We work in the basis where the charged lepton mass matrix is diagonal. The most general neutrino mass matrix M is Majorana in nature, and may be diagonalized by a complex orthogonal transformation into a real diagonal matrix M D = U T MU, () by a unitary matrix U, which is generally obtained from the Hermitian matrix M M = M M by M DM D = U M MU [4]. Some general properties of Majorana neutrino mass matrices are discussed in more detail in Appendix A. For the four-neutrino case, labeling the flavor eigenstates by ν x,ν e,ν µ,ν y and the mass eigenstates by ν 0,ν 1,ν,ν 3, we may write 3

4 ν x ν 0 ν e ν µ = U ν 1 ν. (3) ν y ν 3 In this paper we will examine the two cases most often considered in the recent literature: one of ν x and ν y is ν τ and the other is sterile (ν s ), or both ν x and ν y are sterile. Explicitly, the matrix M may be written in the flavor basis as M xx M xe M xµ M xy M M = xe M ee M eµ M ey M xµ M eµ M µµ M µy. (4) M xy M ey M µy M yy The 4 4 unitary matrix U may be parametrized by 6 rotation angles and 6 phases, and can be conveniently represented by [5] where with U = R 3 R 13 R 03 R 1 R 0 R 01, (5) c 01 s s R 01 = 01 c , (6) c jk cos θ jk, s jk sin θ jk e iδ jk, (7) and the other R jk are defined similarly for rotations in the j k plane. The explicit form for the 4 4 unitary matrix is c 01 c 0 c 03 c 0 c 03 s 01 c 03 s 0 s 03 c 01 c 0 s 03 s 13 c 0 s 01s 03 s 13 s 0s 03 s 13 c 03 s 13 c 01 c 13 s 0 s 1 c 13 s 01 s 0s 1 +c 0 c 13 s 1 c 1 c 13 s 01 +c 01 c 1 c 13 c 01 c 0 c 13 s 03 s 3 c 0c 13 s 01 s 03s 3 c 13s 0 s 03s 3 c 03c 13 s 3 +c 01 s 0 s 1s 13 s 3 +s 01s 0 s 1s 13 s 3 c 0 s 1s 13 s 3 c U = 01 c 1 c 3 s 0 c 1 c 3 s 01 s 0 +c 0 c 1 c 3 +c 1 s 01 s 13 s 3 c 01 c 1 s 13 s. (8) 3 +c 3 s 01 s 1 c 01 c 3 s 1 c 01 c 0 c 13 c 3 s 03 c 0 c 13 c 3 s 01 s 03 c 13 c 3 s 0 s 03 c 03 c 13 c 3 +c 01 c 3 s 0 s 1s 13 +c 3 s 01s 0 s 1s 13 c 0 c 3 s 1s 13 +c 01 c 1 s 0 s 3 +c 1 s 01 s 0s 3 c 0 c 1 s 3 +c 1 c 3 s 01 s 13 c 01 c 1 c 3 s 13 s 01 s 1 s 3 +c 01 s 1 s 3 4

5 We will label the matrix elements of U by U αj, where Greek indices denote flavor eigenstate labels (α = x, e, µ, y) and Latin indices denote mass eigenstate labels (j =0,1,,3). With the knowledge of the mass eigenvalues and mixing matrix elements, one can invert Eq. () to obtain the neutrino mass matrix elements 3 M αβ = Uαj U βj m j. (9) j=0 The vacuum neutrino flavor oscillation probabilities, for an initially produced ν α to a finally detected ν β, can be written where P (ν α ν β )=δ αβ j<k [ 4Re(W jk αβ )sin kj Im(W jk αβ )sin kj], (10) W jk αβ U αjuαku βju βk, (11) kj δm kjl/(4e), δm kj m k m j, (1) L is the oscillation distance, and E is the neutrino energy. The quantities W jk αβ related to the Jarlskog invariants [7] which satisfy the identity obtained by the interchange U U in Eq. (11). Also, We can also define the real part of W jk αβ as [6], are J jk jk αβ Im(Wαβ ), (13) J jk jk αβ = Jαβ, (14) J jk αβ = J kj βα = Jjk βα = Jkj αβ. (15) Y jk αβ Re(W jk αβ ), (16) which is invariant under interchange of the upper or lower indices: Another useful property of the W jk αβ a real positive quantity, e.g., β Equations (10), (14), (15) and (17) imply β Y jk αβ = Y kj βα = Y jk βα = Y kj αβ. (17) is that the sum over any of the indices reduces them to W jk αβ = U αj δ jk = β Y jk αβ, (18) J jk αβ =0. (19) 5

6 P (ν α ν β )=P( ν β ν α ), (0) which is a statement of CPT invariance. Equation (10) and (15) imply that nonzero J jk αβ can give CP or T violation P (ν α ν β ) P ( ν α ν β )=P(ν β ν α ). (1) From Eq. (1) we can define the CP-violation quantity P αβ = P (ν α ν β ) P (ν β ν α ). () In four-neutrino oscillations there are only three independent P αβ, and, correspondingly, only three of the six phases in U can be measured in neutrino oscillations (for a discussion, see Appendix A). Thus six angles and three phases can in principle be measured in neutrino oscillations, which is the same as in the Dirac neutrino case. Therefore, as far as neutrino oscillations are concerned, our results apply equally to Dirac neutrinos. The three remaining independent phases in U enter into the mass matrix elements and processes such as neutrinoless double beta decay. B. Model with two nearly degenerate pairs of neutrinos For a four-neutrino model to describe the solar, atmospheric, LSND data and also satisfy all other accelerator and reactor limits, it must have two pairs of nearly degenerate mass eigenstates [13,14]; e.g., δm sun δm 01 δm atm δm 3 δm LSND δm 1. We will also assume without loss of generality that 0 <m 0,m 1 <m <m 3. An alternative scenario with the roles of δm 01 and δm 3 reversed gives the same results as far as oscillations are concerned, although the implications for the mass matrix, double beta decay and cosmology may differ; this alternate possibility will be briefly discussed in Sec. IV.E. Also note that if the solar oscillation are driven by the MSW effect [8], we must require m 0 >m 1 ;for vacuum oscillations, m 0 <m 1 is also possible. Given this hierarchy of the δm, the oscillation probabilities for α β may be written approximately as P (ν α ν β ) A αβ LSND sin LSND + A αβ atm sin atm + B αβ atm sin atm +A αβ sun sin sun + Bsun αβ sin sun, α β, (3) and for the diagonal channels P (ν α ν α ) 1 A αα LSND sin LSND A αα atm sin atm A αα sun sin sun, (4) where scale 1 4 δm scalel/e, A αβ scale is the usual CP conserving oscillation amplitude for ν α ν β at a given oscillation scale, B αβ scale is the CP violation parameter at a given scale, and the scale label is sun for the solar neutrino scale, atm for atmospheric and long-baseline scales, and LSND for accelerator and short-baseline scales. In Eqs. (3) and (4) we have used the approximation LSND. The oscillation amplitudes are given by 6

7 A αβ LSND =4 U αuβ +U α3uβ3 =4 U α0 Uβ0 +U α1uβ1, α β, (5) A αα LSND =4( U α + U α3 )(1 U α U α3 ), =4( U α0 + U α1 )(1 U α0 U α1 ), (6) A αβ atm = 4 Re(U α Uα3U βu β3 ), α β, (7) A αα atm =4 U α U α3, (8) A αβ sun = 4 Re(U α0 Uα1U β0u β1 ), α β, (9) A αα sun =4 U α0 U α1, (30) where the second equality in Eqs. (5) and (6) follows from the unitarity of U. Wenotethat the form of the short-baseline oscillation amplitudes in Eqs. (5) and (6) are different from the cases of long-baseline, Eqs. (7) and (8), and solar, Eqs. (9) and (30). The difference is due to the fact that the short-baseline oscillations arise from four mass-squared differences (δm 0 δm 30 δm 1 δm 31 ), while the long-baseline and solar oscillations arise from only one mass-squared difference (δm 3 and δm 01, respectively). Probability conservation implies A αα scale = β α A αβ scale, which can easily be shown using the unitarity of U. TheCP violation parameters are Since B αα j B αβ atm = Im(U α Uα3U βu β3 ), (31) Bsun αβ =Im(U α0 Uα1U β0u β1 ). (3) = 0, there is no CP violation in diagonal channels. The absence of B αβ LSND in Eq. (3) shows that no observable CP violation is present for the leading oscillation [9], and CP violation may only be seen in experiments that probe non-leading scales, δm atm or δm sun. For short-baseline experiments where only the leading oscillation argument LSND has had a chance to develop, the off-diagonal vacuum oscillation probabilities are P (ν α ν β ) A αβ LSND sin LSND, α β, (33) P (ν α ν α ) 1 A αα LSND sin LSND. (34) For larger L/E (such as in atmospheric and long-baseline experiments), where the secondary oscillation has had time to develop, the vacuum oscillation probabilities are P (ν α ν β ) 1 Aαβ LSND + A αβ atm sin atm + B αβ atm sin atm, α β, (35) P (ν α ν α ) 1 1 Aαα LSND A αα atm sin atm. (36) Here we have assumed that the leading oscillation has averaged, i.e., sin LSND 1. Finally, at the solar distance scale, when all oscillation effects have developed, the vacuum oscillation probabilities are P (ν α ν β ) 1 (Aαβ LSND + A αβ atm)+a αβ sun sin sun + B αβ sun sin sun, α β, (37) P (ν α ν α ) 1 1 (Aαα LSND + Aαα atm ) Aαα sun sin sun, (38) 7

8 where sin atm has been averaged to 1 and sin atm has been averaged to 0; the CP violation at the δm atm scale is washed out. CP violation is possible only in the off-diagonal channels, as noted before, and the solar neutrino ν e survival measurement cannot be used to observed CP violation. More generally, in any model for which the oscillation scales are well-separated and L/E is only large enough to probe the largest oscillation scale, CP-violating effects in neutrino oscillations will be unobservable (strictly speaking, they are suppressed to order 1, where is the oscillation argument for the second-largest oscillation scale) [9]. The CPviolating effects become observable when L/E is large enough to probe both the largest and second-largest oscillation scales. For a four-neutrino model with different oscillation scales to describe the solar, atmospheric and LSND data, this means that CP violation can only be detected in experiments with L/E at least as large as those found in atmospheric and long-baseline experiments. As a corollary, in three-neutrino models with two oscillation scales describing only the solar and atmospheric data, CP violation has the potential to be observable only in experiments with L/E comparable to or larger than the solar experiments. However, a measurement of off-diagonal oscillation probabilities is required to see CP violation, and that is not possible in solar neutrino experiments. Hence, if the solar neutrino oscillation scale is well-established, the observation of a CP-violation effect in long-baseline experiments could imply that there are at least three separate neutrino mass-squared difference scales, and thus more than three neutrinos. III. DETERMINING THE OSCILLATION PARAMETERS In this section we first derive some general constraints imposed by current data on the neutrino mixing matrix for four-neutrino models favored by the data, i.e., with two pairs of nearly degenerate masses satisfying δm 01 δm 3 δm 1. We then determine the form of the mixing matrix under the assumption that ν e and ν x are mostly a mixture of ν 0 and ν 1 and provide the dominant solar neutrino oscillation, and ν µ and ν y are mostly a mixture of ν and ν 3 and provide the dominant atmospheric neutrino oscillation, which is the form of most explicit models in the literature. Then we discuss the measurements needed to determine the parameters in the neutrino mixing matrix. The results of this section apply equally to the case where m <m 3 <m 0,m 1. The more general case where ν e and ν µ have large mixing with more than one other neutrino is briefly discussed in Sec. IV.E. A. Solar ν e ν x and atmospheric ν µ ν y The flavor eigenstates are related to the mass eigenstates by Eq. (3). Using the formulae in Sec. II, the amplitudes for short-baseline oscillation (such as LSND, reactors, and other past accelerator oscillation searches) are A eµ LSND =4 U e U µ+u e3 U µ3 =4 U e0 U µ0+u e1 U µ1, (39) A µµ LSND =4( U µ + U µ3 )(1 U µ U µ3 ) =4( U µ0 + U µ1 )(1 U µ0 U µ1 ), (40) 8

9 A ee LSND =4( U e + U e3 )(1 U e U e3 ) =4( U e0 + U e1 )(1 U e0 U e1 ), (41) where the second equalities in each case result from the unitarity of U. For atmospheric and long-baseline oscillation, the amplitudes are and in solar experiments A µµ atm =4 U µ U µ3, (4) A eµ atm = 4 Re(U e Ue3 U µ U µ3), (43) B eµ atm = Im(U e Ue3 U µ U µ3), (44) A ey atm = 4 Re(U e Ue3 U y U y3), (45) B ey atm = Im(U e Ue3U yu y3 ), (46) A µy atm = 4 Re(U µ Uµ3 U y U y3), (47) B µy atm = Im(U µ Uµ3 U y U y3), (48) A ee sun =4 U e0 U e1. (49) The atmospheric neutrino experiments favor large mixing of ν µ at the atmospheric scale [14,30] at 90% C.L.; then Eq. (4) and unitarity imply A µµ atm > 0.8, (50) U µ + U µ3 > 0.894, U µ0 + U µ1 < (51) Also, the Bugey reactor constraint [31] gives over the indicated range for δm LSND; then Eq. (41) implies A ee LSND < 0.06, (5) U e + U e3 < 0.016, U e0 + U e1 > (53) Finally, the oscillation interpretation of the LSND results [9] gives where ɛ is experimentally constrained to the range [9] A eµ LSND ɛ, (54) 0.05 <ɛ<0.0, (55) where the exact value depends on δm LSND. Hence, the atmospheric and Bugey results imply that U e, U e3, U µ0,and U µ1 are all approximately of order ɛ or smaller. We note that given these constraints the size of A µµ LSND must also be small, in agreement with the CDHS bound on ν µ disappearance [3]. 9

10 If we assume that it is only ν y that mixes appreciably with ν µ in the atmospheric experiments and only ν x that mixes appreciably with ν e in the solar experiments, then U x, U x3, U y0,and U y1 must also be small, i.e., of order ɛ or less. The mixing matrix can therefore be seen to have the form ( ) U1 U U =, (56) U 3 U 4 where the U j (j = 1,,3,4) are matrices and the elements of U and U 3 are at most of order ɛ in size. The matrix U 4 is approximately the maximal mixing matrix (i.e., all elements have approximate magnitude 1/ ) that describes atmospheric ν µ ν y oscillations, but U 1, which is approximately unitary by itself and which primarily describes the mixing in the solar neutrino sector, may have large (for vacuum oscillations) or small (for MSW oscillations) mixing. The form of U in Eq. (56), with U and U 3 ɛ, implies s 0, s 03, s 1, s 13 ɛ, (57) in the general parametrization of Eq. (8). After dropping terms second order in ɛ and smaller, U takes the form c 01 s 01 s 0 s 03 s 01 c 01 s 1 s 13 U c 01 (s 3s 03 + c 3 s 0 ) s 01(s 3s 03 + c 3 s 0 ) c 3 s 3 +s 01 (s 3 s 13 + c 3 s 1 ) c 01 (s 3 s 13 + c 3 s 1 ). (58) c 01 (s 3 s 0 c 3 s 03 ) s 01(s 3 s 0 c 3 s 03 ) s 3 c 3 s 01 (s 3 s 1 c 3 s 13 ) +c 01 (s 3 s 1 c 3 s 13 ) This matrix provides a general parametrization of the four-neutrino mixing in models where ν e mixes primarily with ν x at the solar mass-squared difference scale, and ν µ mixes primarily with ν y at the atmospheric mass-squared difference scale. Unitarity of U is satisfied to the order of ɛ. Despite the fact that the expansion of the matrix elements of U in Eq. (58) is to the first order of ɛ, it still allows us, as shown below, to extract all of the interesting oscillation and CP violation effects, which are second order in ɛ. Care must be taken when the leading order result cancels, and sometimes it is helpful to use the unitarity of U to derive an alternate expression that gives the correct leading order answer, e.g., for A µy LSND, the first expression in Eq. (5) gives zero when the form of U in Eq. (58) is used, but the second expression gives a finite (and correct to leading order) result. The off-diagonal oscillation amplitudes for the leading oscillation are A eµ LSND =4 s 1c 3 + s 13 s 3, (59) A ey LSND =4 s 1 s 3 s 13 c 3, (60) A µx LSND =4 s 0 c 3 + s 03 s 3, (61) A ex LSND = Aµy LSND = O(ɛ4 ). (6) 10

11 For atmospheric and long-baseline experiments the oscillation amplitudes are A eµ atm = A ey atm = 4c 3 Re(s 1 s 13s 3 ), (63) B eµ atm = B ey atm = c 3 Im(s 1s 13 s 3), (64) A ex atm,batm ex = O(ɛ 4 ), (65) A µµ atm A µy atm =sin θ 3, (66) A µx atm = 4c 3 Re(s 0 s 03s 3 ), (67) B µx atm =c 3 Im(s 0s 03 s 3), (68) B µy atm = c 3 Im[(s 0 s 03 + s 1 s 13)s 3 ], (69) where θ 3 is defined in Eq. (7). All oscillation amplitudes at the LSND scale and all oscillation amplitudes (including the CP violation amplitudes) other than A µy atm at the atmospheric scale are at most of order ɛ. Since A µy atm is large, B atm, µy which is of order ɛ,maybehard to measure since it involves taking the difference of two nearly equal large numbers. At the solar scale, we have A ee sun Aex sun =sin θ 01. (70) There are 6 mixing angles and 3 independent phases that in principle may be measured in neutrino oscillations. In all, there are eight independent parameters involved in the observables in Eqs. (59) (70), which are the six mixing angles θ 01, θ 0, θ 03, θ 1, θ 13, θ 3,and the two phases φ 0 δ 03 δ 0 δ 3, (71) φ 1 δ 13 δ 1 δ 3. (7) These eight parameters could in principle be determined by measurements of the eight observables A ee sun, Aµµ atm, A eµ LSND, Aeµ atm, Batm, eµ A µx LSND, Aµx atm, andb atm. µx Therefore if ν x = ν τ, then all eight of these parameters could in principle be determined from the solar, atmospheric, short- and long-baseline experiments. This emphasizes the need for both shortand long-baseline measurements of all active oscillation channels, since the oscillation amplitudes involve different combinations of the parameters at short and long baselines. If ν x is sterile, the three parameters θ 0, θ 03 and φ 0 might be difficult to determine since they would involve the disappearance ν µ ν s that is at most of order ɛ in magnitude. If ν y = ν τ, the additional observables A ey LSND, Aey atm, andb ey atm can provide a consistency check on the parameters θ 1, θ 13,andφ 1. We note that from the above results that many of the CP-violating amplitudes can be the same order of magnitude as the corresponding CP-conserving amplitudes, and hence potentially observable in high-statistics long-baseline experiments. The CP violation parameters B eµ atm and B µx atm could be determined in vacuum by measuring probability differences P eµ and P µx,where P αβ P (ν α ν β ) P ( ν α ν β ), (73) in long-baseline experiments, or the probability differences P eµ and P µx,where P αβ, defined in Eq. (), measures explicit T -violation. In a vacuum, 11

12 Alternatively, one could measure CP asymmetries or T asymmetries P αβ = P αβ =B αβ atm sin atm. (74) A CP αβ = P (ν α ν β ) P ( ν α ν β ) P(ν α ν β )+P( ν α ν β ), (75) A T αβ = P (ν α ν β ) P (ν β ν α ) P (ν α ν β )+P(ν β ν α ). (76) In vacuum, CPT invariance insures that P αβ = P αβ and A T αβ = ACP αβ. However, matter effects could induce a nonzero P αβ or A CP αβ even in the absence of CP violation [18,33]. Since the matter effects in long-baseline experiments for P (ν α ν β )andp(ν β ν α )are the same, the quantities P eµ and A T eµ, which can only be nonzero if there is explicit CP or T violation, may be preferable [18]. There remains a third independent phase that could have consequences for neutrino oscillations, but will in practice be difficult to measure. This phase, which could be taken as δ 01 defined in Eq. (5), could be determined from CP violation in ν e ν µ or ν y at the δm sun scale, but this effect would require the measurement of an off-diagonal channel at the solar scale. Therefore it appears that a complete determination of the four-neutrino mixing matrix is not possible with conventional oscillation experiments. Table I lists all parameters that appear to be accessible to observation together with the principal observables that determine these parameters. B. More general mixing scenarios In general, both solar ν e and atmospheric ν µ could oscillate into mixtures of ν x and ν y. In this event θ 0 and θ 03 in Eq. (8) are not necessarily small. If one of ν x and ν y is the tau neutrino and the other sterile, there are several possible ways that the existence of such mixing could be determined [34]. Also, vacuum CP-violation effects involving ν e will still be no larger than order ɛ (due to the smallness of U e and U e3 ), but there are potentially large CP-violation effects in long-baseline ν µ -ν y oscillations (as large as allowed by the unitarity of U) [18]. IV. CP VIOLATION AND NEUTRINO MASS TEXTURES In this section we study the relationship between the neutrino mass texture and the possibility for observable CP violation (and, equivalently, T violation) in neutrino oscillations in four-neutrino models. We will consider models where one of ν x and ν y is sterile and the other is ν τ (such as in Refs. [14], [19], [0], and [35]), and also models where both are sterile [36], which are two possible extensions of the Standard Model neutrinos. Note that in all earlier studies the mass matrices were taken to be real and no CP violation was possible. In Appendix B we discuss straightforward extensions of the Standard Model for the two 1

13 cases and show explicitly how their neutrino mass matrices can arise. In Secs. IV.A IV.D we assume that m 0 m 1 m m 3 δm LSND, (77) i.e., the lighter pair of nearly-degenerate mass eigenstates are much lighter than the heavier pair, also nearly degenerate, which is the structure of most explicit four-neutrino models in the literature. In Sec. IV.E we briefly discuss models with other mass hierarchies. From Eqs. (9) and (58) the neutrino mass matrix elements involving ν µ and ν y are, to leading order in ɛ, M µµ Myy m(c 3 + s 3 ), (78) M µy im sin θ 3 sin δ 3, (79) M eµ m(s 1 c 3 + s 13 s 3 ), (80) M ey m(s 13 c 3 s 1 s 3 ), (81) M xµ m(s 0 c 3 + s 03 s 3 ), (8) M xy m(s 03 c 3 s 0 s 3), (83) wherewehaveusedtherelativesizesoftheu αj and mass eigenvalues, and the fact that to leading order in ɛ, m m 3 m. Note that the s jk, defined in Eq. (7), may be complex. For the mass matrix elements M xx, M xe,andm ee, all four terms in Eq. (9) are small and may be of similar size (the first two are suppressed by the small values of m 0 and m 1,the last two by mixing angles of size ɛ); their values depend on the exact structure in the solar sector, which we do not specify here. More precise solar neutrino measurements would help to determine their values. The three phases which enter in the expressions for the mass matrix elements given above are δ 3 and φ 0 δ 03 δ 0 + δ 3, (84) φ 1 δ 13 δ 1 + δ 3. (85) Only the phases φ 0 and φ 1, which can be measured in oscillation experiments, and φ 0,which appears in the expressions for the mass matrix elements, are independent. The two phases δ 3 =(φ 0 φ 0)/andφ 1 =φ 0 +φ 1 φ 0 are linearly dependent. Equations (78) (83) may be used to examine the implications of specific textures of the neutrino mass matrix. In the following, we discuss several specific textures of the neutrino mass matrix which have been considered in the literature. Their CP effects are particularly noted. A. M eµ =0 Specific examples of this class of models are given in Refs. [14], [19], and [0], in which ν x = ν s, ν y = ν τ, and the mass matrices are taken to be real. In these models the mass matrices were chosen to minimize the number of parameters needed to provide the appropriate phenomenology, and a nonzero M eµ is not required. In Ref. [14] the case ν x = ν τ and ν y = ν s was also considered. 13

14 Using Eq. (80), M eµ = 0 implies s 1 c 3 s 13 s 3, which in turn leads to A eµ LSND 16 s 13 s 3 sin δ 3, (86) A ey LSND 4 s 13 (1 sin θ 3 sin δ 3 )/c 3, (87) A eµ atm = A ey atm 4 s 13 s 3 cos δ 3, (88) B eµ atm = B ey atm s 13 s 3 sin δ 3, (89) and φ 0 = δ 3. Observable oscillations at LSND requires θ 3 0,π and δ 3 0,π. Also, ν e ν y oscillations in short-baseline experiments, ν e ν µ and ν e ν y oscillations in longbaseline experiments, and CP violation in long-baseline experiments are possible, although not required, in this scenario. Finally, we have M µµ M yy m 1 sin θ 3 sin δ 3, (90) M µy msin θ 3 sin δ 3. (91) Asoneexample,ifwetakeδ 3 π and θ 3 π,weobtainthemodelofref.[14],in 4 which there is maximal ν µ -ν y mixing and M µµ M yy M µy. Furthermore A ey LSND = B eµ atm = B ey atm =0,soν e oscillates only to ν µ in short-baseline experiments and there is no visible CP violation in long-baseline experiments. If we allow θ 3 π, we have a model equivalent to that of Ref. [19]; in this case Aey 4 LSND 0 and B eµ atm = B ey atm =0,sotherecanbeν e ν y oscillations in short-baseline experiments but still no visible CP violation in long-baseline experiments. In order to have CP violation in the present case, we must have δ 3 π. Then there must be short-baseline ν e ν y oscillations, although the existence of long-baseline ν e ν µ oscillations depends on the value of θ 3. Finally, an interesting case to consider is maximal CP violation (maximal in the sense that it gives the largest CP-violation parameter for a given s 13 and s 3 ), which corresponds to δ 3 = π. If there is also maximal ν 4 µ-ν y mixing (θ 3 = π ), then the mass matrix in the ν 4 µ-ν y sector is approximately (after appropriate changes of neutrino phase to make the diagonal elements real to leading order in ɛ) ( ) Mµµ M µy m ( ) ( ) 1 i + δm atm 1 1 M yµ M yy i 1 4m eiπ/4. (9) 1 1 The measurables in short- and long-baseline experiments are then A eµ LSND = A ey LSND 4 s 13, (93) A eµ atm = A ey atm 0, (94) B eµ atm = B ey atm s 13, (95) i.e., ν e oscillates equally into ν µ and ν y in short-baseline experiments and there are no additional contributions to the CP-conserving part of these oscillations in long-baseline experiments. The vacuum CP and T asymmetries are especially simple in this case, A CP eµ = A T eµ = A T ey 1 sin atm, (96) as the dependence on s 13 cancels in the ratio. The particular models discussed above are summarized in Table II. 14

15 B. M ey =0 An example of this class of models with ν x = ν s and ν y = ν τ is given in Ref. [35], where a nonzero M ey was not needed to provide the appropriate phenomenology. From Eq. (81), M ey = 0 implies s 1 s 3 s 13c 3, which leads to A eµ LSND 4 s 13 (1 sin θ 3 sin δ 3 )/ s 3, (97) A ey LSND 16 s 13 c 3 sin δ 3, (98) A eµ atm = A ey atm 4 s 13 c 3 cos δ 3, (99) B eµ atm = B ey atm s 13 c 3 sin δ 3, (100) and φ 0 = δ 3. The existence of oscillations in LSND implies that δ 3 π or θ 3 π 4.As with the M eµ = 0 case, it is possible to have ν e ν y oscillations in short-baseline experiments, ν e ν µ and ν e ν y oscillations in long-baseline experiments, and CP violation in long-baseline experiments. The approximate magnitudes of the mass matrix elements M µµ, M yy,andm µy are the same as given in Eqs. (90) and (91). The limit δ 3 0andθ 3 π 4 reproduces the model in Ref. [35], which has no ν e ν y in short-baseline experiments and no visible CP violation in long-baseline experiments. CP violation can occur if δ 3 0, π,π, in which case there are ν e ν y oscillations in shortbaseline experiments and there may be ν e ν µ and ν e ν y oscillations in long-baseline experiments, depending on the value of θ 3.TheM ey = 0 model with maximal CP violation and maximal ν µ -ν y mixing (δ 3 = θ 3 = π 4 ) has the same features as the M eµ = 0 maximal CP violation case in Sec. IV.A, except that the vacuum CP and T asymmetries in Eq. (96) have the opposite sign. The particular M ey = 0 cases discussed here are also summarized in Table II. C. M eµ 0and M ey 0 In this more general case, barring fortuitous cancellations one would expect from Eqs. (59) (68) that there are ν e ν µ and ν e ν y oscillations in short- and long-baseline experiments, and CP violation in ν e ν µ and ν e ν y oscillations in long-baseline experiments. For this texture, M µµ, M yy M µy does not necessarily exclude visible CP violation, unlike the cases M eµ =0orM ey = 0. As an example, the mass matrix ɛ 1 ɛ e iφ 0 0 ɛ M = m e iφ 0 ɛ 5 ɛ 3 e iφ 3 0 ɛ 5 ɛ 4 e iφ 1 0 ɛ 3 e iφ 3 e iφ 1 ɛ 6, (101) which is an extension of the model introduced in Ref. [14], leads to CP violation of order ɛ in ν e ν µ and ν e ν y oscillations in long-baseline experiments. This mass matrix differs from the one in Ref. [14] in that the M eµ and M µe elements, denoted as ɛ 5, are not zero, the M µµ and M yy elements, denoted as ɛ 4 and ɛ 6, respectively, are not necessarily equal, and the CP-violating phases are not set to zero. The diagonal elements of the mass matrix can be 15

16 taken to be real. Because M ee = M xµ = M xy = 0 there are only three independent phases. The mass eigenvalues and approximate mixing matrix for the mass matrix in Eq. (101) are given in Appendix C. The largest off-diagonal short-baseline oscillation amplitudes in this case are A eµ LSND 4ɛ 3, (10) A ey LSND 4ɛ 5. (103) Short-baseline ν µ -ν y oscillations are of order ɛ 4. The largest long-baseline oscillation probabilities are A µy atm 1, (104) A eµ atm = A ey atm ɛ 5 ɛ 3. (105) Short- and long-baseline oscillation amplitudes involving ν x are of order ɛ 4 or smaller. The observable CP-violating amplitude is where (see Appendix C) Finally, B eµ atm ɛ 3 ɛ 5 sin(φ 3 + δ 3 ), (106) tan δ 3 = (ɛ 4 ɛ 6 )sinφ 1 +ɛ 3 ɛ 5 sin φ 3 (ɛ 4 + ɛ 6 )cosφ 1 +ɛ 3 ɛ 5 cos φ 3. (107) A ee sun =sin θ 01, (108) where tan θ 01 = ɛ 1+ 4ɛ 1ɛ 3 ɛ 5 (1 cos(φ 1 + φ 3 φ )). (109) ɛ 1 +ɛ 3 ɛ 5 (ɛ 1 ɛ 3 ɛ 5 ) The phenomenology of these models is summarized in Table II. D. M ee = M eµ = M µµ =0 In the special class of models with two sterile and two active neutrinos, i.e., both ν x and ν y are sterile, there need not be Majorana mass terms for the two active neutrinos in order to obtain the proper phenomenology. As described in more detail in Appendix B, M ee = M eµ = M µµ = 0 requires only a minimal extension in the Higgs sector of the Standard Model, i.e., only SU() singlet Higgs bosons need to be added. Examples of models with both ν x and ν y sterile are given in Ref. [36]. We will now show that CP violation effects in long-baseline experiments are suppressed in this class of models. Models with M eµ = 0 have already been discussed in Sec. IV.A; here we add the additional constraint M µµ = 0. It was previously determined that M µµ, M yy M µy implied δ 3 π and θ 3 π, so Eqs. (86) (89) reduce to 4 16

17 A eµ LSND 8 s 13, (110) A ey LSND 0, (111) A eµ atm = A ey atm s 13, (11) B eµ atm = B ey atm 0. (113) Hence, there is no visible CP violation in long-baseline experiments in this case (strictly speaking CP violation is strongly suppressed, to order ɛ 4 ). Therefore in models with M ee = M eµ = M µµ = 0, the only phenomenological deviations from the Standard Model are CPconserving neutrino oscillations (see Table II) and the presence of more than one neutral Higgs scalar. E. Other mass hierarchies There are other mass hierarchies possible which give the same oscillation phenomena as those discussed in Secs. IV.A IV.D. One is m <m 3 m 0,m 1 δm LSND m, inwhich the solar oscillation occurs between the two upper mass eigenstates and the atmospheric oscillation between the two lower mass eigenstates. Assuming as before that ν e mixes primarily with ν x and ν µ with ν y (i.e., s 0,s 03,s 1,s 13 ɛ), then M eµ m(u e0 U µ0 + U e1 U µ1 ) ; (114) M ee, M ex, M xx, M ey, M xµ, and M xy are given by similar expressions with appropriate changes of subscripts. However, for M µµ, M µy,andm yy none of the four terms in Eq. (9) are dominant. Since long-baseline ν α ν β oscillations involve δm 3, and hence the mixing matrix elements U α, U α3, U β,andu β3 (see Eqs. (4) (48)), then a mass texture condition such as M eµ = 0, when applied to Eq. (114), does not tell us anything specific about longbaseline oscillations. It could, however, affect short-baseline oscillation amplitudes, which depend on the U α0 and U α1 (see Eqs. (39) (41)). We do not pursue this possibility further here. Another possible hierarchy is to have m 0 m 1 <m m 3 where none of the masses are much smaller than the others; in this case, all masses would contribute to hot dark matter (an alternate possibility, m m 3 <m 0 m 1 with none small, gives similar results). A model of this type has been discussed in Ref. [0]. In this case, again assuming s 0,s 03,s 1,s 13 ɛ, we find with similar expressions for M ee and M xx,and M ex m 0 (U e0u x0 + U e1u x1), (115) M µy m 3 (U µu y + U µ3u y3), (116) with similar expressions for M µµ and M yy. However, for M eµ, M ey, M xµ,andm xy,noneof the four terms in Eq. (9) are dominant, the expressions for the mass matrix elements are more complicated, and the implications of particular textures for long-baseline oscillations are not as easily determined. We also do not pursue this case further here. 17

18 V. OTHER PROBES OF NEUTRINO MASS The presence of mass terms for neutrinos and, in particular, Majorana mass terms, opens up a variety of possibilities for phenomena that are not possible in the Standard Model. The neutrino mass, besides giving rise to mixing of neutrinos and the associated CP effect discussed in this paper, can lead to lepton flavor-changing charged currents analogous to those of the quark sector. Majorana mass can also give rise to lepton number violation processes. With these possibilities, widely searched-for phenomena such as µ e + γ, µ e +ē+e, µ-econversion, and electric dipole moments for charge leptons, can occur. Unfortunately, all these processes [37] are proportional to (m ν /M W ) 4 or ((m ν /M W )ln(m ν /M W ))4.Giventhat m ν is of the order of 5 ev or less [38] these are no larger than and The current upper bounds [39] are about 30 orders of magnitude larger than those theoretical predictions from the neutrino masses. Therefore they are unobservable. Since this conclusion depends only on the smallness of the neutrino masses, it is valid in general. The Majorana mass term breaks lepton number conservation and can lead to neutrinoless double beta decay. The rate is governed by the magnitude of the effective ν e mass m νe = j U ej m j = M ee, (117) i.e., the magnitude of the M ee element in the Majorana neutrino mass matrix in Eq. (4). The current limit on M ee from neutrinoless double beta decays is about 0.5 ev [40]. Since M ee = s 01 m 0 + c 01 m 1 + s 1 m + s 13 m 3, (118) there will be no visible neutrinoless double beta decay in models with m 0,m 1 m <m 3 δm LSND 1eVand s 1 s 13 ɛ.inmodelswherem <m 3 m 0,m 1 δm LSND, or if no neutrino masses are 1 ev, neutrinoless double beta decay may provide a strong constraint. Neutrino masses may also affect cosmology if ν m ν > 0.5 ev [41]. This level of neutrino mass can easily be accommodated by a four-neutrino model with two pairs of nearly degenerate mass eigenstates separated by approximately 1 ev. Because of the smallness of the neutrino masses, there are no other observable effects besides neutrino oscillations and possibly neutrinoless double beta decay and dark matter. However, the rare decays may still be observable if new physics occurs also in other sectors, such as anomalous gauge boson couplings or anomalous fermion-gauge boson interactions. Therefore, it is important to continue to search for them. VI. SUMMARY In this paper we have presented a general parametrization of the four-neutrino mixing matrix and discussed the oscillation phenomenology for the case of two nearly degenerate pairs of mass eigenstates separated from each other by approximately 1 ev, which is the mass spectrum indicated by current solar, atmospheric, reactor, and accelerator neutrino experiments. We analyzed in detail the case where ν e mixes primarily with ν x and ν µ with ν y,whereoneofν x and ν y is ν τ and the other is sterile, or both are sterile. We found in these 18

19 cases that the neutrino mixing matrix can be written in block form with small offdiagonal blocks. By construction the mixing matrices have ν e ν µ oscillations in LSND and ν µ ν y oscillations in atmospheric experiments. We found that the following oscillations are also possible: ν e ν y and ν µ ν x in short-baseline experiments, and ν e ν µ, ν e ν y,and ν µ ν x (including CP-violation effects) in long-baseline experiments. We also found that solar, atmospheric, short- and long-baseline oscillation measurements can, in some cases, determine all but one of the four-neutrino mixing parameters. Finally, we examined the implications of some several specific mass textures, and found the conditions under which CP-violation effects are visible. As pointed out in the Introduction, additional evidence is needed in order for the fourneutrino scenario to be on firm ground. Given that there must be separate mass-squared difference scales for the solar and atmospheric oscillations (as currently indicated by the data), there are in fact two ways to verify the existence of four light neutrinos: (i) confirmation of the LSND results, which could occur in the future mini-boone collaboration [4 44], or (ii) detection of vacuum (i.e., not matter-induced) CP or T violation in longbaseline experiments, which should be greatly suppressed in a three-neutrino scenario. Once the existence of four neutrinos is established, the next task is to determine the neutrino mixing matrix parameters. We emphasize the significant potential for detecting new oscillation channels and CP violation in future high statistics short- and long-baseline oscillation experiments. Many experiments have been proposed and some will be online in the next few years [44,45]. In these experiments neutrino beams are produced at high energy accelerators and oscillations can be detected at distant underground detectors. They include the KEK-Kamiokande KK Collaboration [46], the Fermilab-Soudan MINOS [47] and Emulsion Sandwich [48] collaborations, and CERN-Gran Sasso ICARUS, Super-ICARUS, AQUA-RICH, NICE, NOE and OPERA collaborations [49]. Experiments done at muon storage rings [45] may be especially important since they will have the ability to measure both ν e ν µ and/or ν τ,andν µ ν e and/or ν τ, as well as the corresponding oscillation channels for antineutrinos. Furthermore, there may also be hitherto undiscovered oscillation effects in short-baseline oscillation experiments such as COSMOS [50] and TOSCA [51], which will search for ν µ ν τ oscillations. To completely determine all accessible parameters in the four-neutrino mixing matrix requires searches at both short and long baselines. The amplitudes of various oscillation channels, including possible CP violation effects, will help further determine the texture of the four-neutrino mass matrix and offer a better understanding of neutrino physics as well as CP violation. ACKNOWLEDGEMENTS YD would like to thank the hospitality of Iowa State University where this work began. This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, under Grants No. DE-FG0-94ER40817 and No. DE-FG0-95ER40896, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. YD s work is partially support by the National Natural Science Foundation of China. BLY acknowledges the support by a NATO collaborative grant. 19

20 APPENDIX A: GENERAL PROPERTIES OF N N MAJORANA MASS AND MIXING MATRICES Consider the general case of n neutrinos, in which n R are right-handed (sterile) and n L are left-handed (active), n R + n L = n. We can represent the n R right-handed neutrinos by their left-handed conjugates and denote the collection of all the n independent left-handed neutrinos by a column vector ψ L. Then either Dirac or Majorana neutrino mass terms can be written as ψ R Mψ L + h.c., (A1) with ψ R related to ψ L by ψ R = ψlc, T wherec=iγ γ 0 is the charge conjugation matrix. The most general n n mass matrix M is symmetric and complex, M = M, characterized by 1 n(n + 1) magnitudes and the same number of phases. Since a given component field in ψ L and ψ R acquires the same phase factor under a change of phase, n of the phases in M may be absorbed into the definitions of the fields, leaving 1 n(n + 1) magnitudes and 1n(n 1) phases; we will often choose the convention that the diagonal elements of M are real and the off-diagonal elements complex. As shown in Eq. (), the mass matrix may be diagonalized by a unitary matrix U. The effects of U may be divided into two classes of phenomenology which give rise to violation of individual lepton number: neutrino oscillations and lepton charged currents. Since Z 0 interacts only with the left-handed neutrinos, neutrino counting in Z ν ν at LEP is unchanged if all of the neutrinos are light, i.e., (N ν ) LEP = n L, with n L =3in the Standard Model. This can be demonstrated straightforwardly as follows. The neutral current Lagrangian can be written in terms of flavor eigenstates as g ψ L γ µ Kψ L Zµ,whereK 0 is an n n diagonal matrix with the first n R elements being zero and the other n L elements unity. If all neutrinos are very light, which is the case we are considering here, the number of neutrinos measured at LEP is simply Tr(KK )=n L, assuming the couplings of the active neutrinos are universal. In terms of the mass eigenstates, the neutrino counting is unchanged: Tr(UK(UK) )=Tr(KK )=n L. In general an n n unitary matrix such as U can be described by 1 n(n 1) rotation angles and 1 n(n + 1) phases. In the charged lepton current, we are free to make phase transformations of the charged lepton fields, which removes n of the phases. Then the number of surviving measurable phases is 1n(n +1) n = 1 n(n 1). This argument is not affected by the fact that the number of left-handed charged lepton fields, n c,inthe charged-current is less than n, as each of the first n n c rows of U can be multiplied by a phase without altering the charged currents. Therefore, in general in this Majorana setting we can parametrize U by 1n(n 1) angles and 1 n(n 1) phases. In neutrino oscillations, however, only 1 (n 1)(n ) independent phases can in principle be measured, as we will now demonstrate. Note that the W jk αβ in Eq. (11) are invariant when U is transformed from either the left or right side by a diagonal matrix which contains only phases, i.e., U αj e iφα U αj e iφ j. (A) Then, as far as neutrino oscillations are concerned, we can eliminate n 1 of the phases in U, so that there are effectively only 0

21 1 n(n +1) (n 1) = 1 (n 1)(n ), (A3) independent phases that can be measured by neutrino oscillation experiments. Interestingly this is the same number of independent phases that may be determined in the CKM matrix for n generations of quarks. However, note that in the most general U there are 1 n(n 1) phases, so that there are 1 n(n 1) 1 (n 1)(n ) = (n 1) phases that cannot be determined from neutrino oscillations. The phase counting of Eq. (A3) can also be confirmed by enumerating the number of independent CP-violating variables, e.g., the P αβ defined in Eq. (), that can be measured. There are 1n(n 1) such differences, but from Eq. (15) P αβ = P βα and from Eq. (19) β P αβ = 0, so it follows that n 1 of the differences are not independent. Therefore there are only 1(n 1)(n ) independent P αβ, and only 1 (n 1)(n ) CP-violation parameters can be measured. APPENDIX B: HIGGS BOSON ORIGINS OF NEUTRINO MASSES The presence of masses for neutrinos is a definite signal of physics beyond the Standard Model. Particularly, with the three types of neutrino oscillations which indicate three δm scales and require at least four neutrino mass values, a non-trivial extension of the Standard Model is necessary. In searching for hints of the extension, it is interesting to consider what simplest extensions of the Standard Model are possible and how natural (or unnatural) they are in their couplings schemes. In this appendix we discuss briefly the possible origins of the two types of mass matrices considered, i.e., models with one or two sterile neutrinos. Then both Dirac (active-sterile) and Majorana (active-active or sterile-sterile) neutrino mass terms are present. These masses can be obtained by suitable extensions of the Standard Model. We will only enlarge the lepton Yukawa sector and the Higgs sector to the extent required by the mass matrices of Sec. IV. Our goal is to assure that such mass matrices are possible by straightforward modifications of these two sectors, and to determine what new particles need to be added to the Standard Model spectrum. We do not attempt to construct the best case scenario, which can be done when more information on the neutrinos mass are available. For a more extensive discussion of possible origins of neutrino masses terms, see Ref. [5]. In the following we denote the right-handed sterile by ν sjr and the corresponding lefthanded conjugate by ˆν sjl = ν sjr T C. We also denote the left-handed lepton SU() doublet by l kl with the corresponding right-handed conjugate fields ˆl kr = iσ C l kr T, j and k are generation labels. 1. Models with two left-handed and two right-handed neutrinos We first consider the class of models in which ν x = ν s1 and ν y = ν s,whereν s1 and ν s are the right-handed sterile neutrinos that are associated with ν e and ν µ, respectively, which have been considered in Ref. [36]. We discuss two cases: (i) models where only the sterile neutrinos have nonzero Majorana mass terms, and (ii) models where both right-handed (sterile) and left-handed (ν e and ν µ ) fields have Majorana mass terms. In the first case 1